Problem 2 - A Graphical Approach to College Algebra

Question

Find the common difference d for each arithmetic sequence. Do not use a calculator. $$4,10,16,22, \dots$$

Step-by-Step Solution

Answer

The common difference \(d\) is 6.

Step 1: Understand the Arithmetic Sequence

Recognize that an arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is known as the common difference, denoted as \(d\).

Step 2: Identify Consecutive Terms

Look at the given sequence: \(4, 10, 16, 22, \dots\). We need to find the common difference by comparing consecutive terms.

Step 3: Calculate the First Common Difference

Subtract the first term from the second term: \(10 - 4 = 6\).

Step 4: Verify Consistency

Check the difference between the next pair of terms to ensure consistency: \(16 - 10 = 6\).

Step 5: Confirm the Common Difference

Subtract the third term from the fourth term to conclude that the difference remains the same: \(22 - 16 = 6\).

Step 6: Conclude the Solution

Since the differences between each pair of consecutive terms are consistent, the common difference \(d\) for the sequence is \(6\).

Key Concepts

Common Difference

An arithmetic sequence is defined by having a consistent interval between each term, known as the "common difference." This is a fundamental trait of arithmetic sequences, and it's crucial to find it in order to understand the characteristics of the sequence. The common difference, usually represented by the letter "d," is what distinguishes an arithmetic sequence from other types of sequences.

It is calculated by taking any term in the sequence and subtracting its preceding term. For instance, in the sequence given by the exercise: 4, 10, 16, 22, ..., we calculate the common difference as follows:

• Subtract the first term from the second term: 10 - 4 = 6
• Verify by subtracting the next term pairs: 16 - 10 = 6 and 22 - 16 = 6

All these calculations give us a result of 6, confirming that the common difference is consistent across the sequence.

Consecutive Terms

Consecutive terms in an arithmetic sequence are pairs of numbers that directly follow one another. Understanding how these terms relate is key to identifying the pattern of the sequence and finding the common difference.

When dealing with a sequence such as 4, 10, 16, 22, ..., consecutive terms play a vital role because:

• They allow for the direct calculation of the common difference.
• They help in verifying the consistency of the sequence.

To explore further, select any two consecutive terms, such as 10 and 16, and find the difference by subtracting these values. This difference should match with the difference from any other pair of consecutive terms in the sequence.

Sequence Consistency

Ensuring sequence consistency is essential to verify that a series of numbers truly forms an arithmetic sequence. This concept means that the common difference is the same between every pair of consecutive terms throughout the sequence.

Consistency can be checked by:

• Calculating the common difference for several pairs of consecutive terms within the sequence.
• Confirming that these differences are equal each time.

In the sequence 4, 10, 16, 22, ..., compare:

• 10 - 4 = 6
• 16 - 10 = 6
• 22 - 16 = 6

Every difference equaling 6 shows sequence consistency, affirming it is an arithmetic sequence.